FOM: Is CH a definite mathematical problem?
sf at Csli.Stanford.EDU
Tue Nov 25 02:10:09 EST 1997
In the previous posting, I gave the reason for my belief that CH is an
inherently vague problem as being based on a "gut feeling". Of course,
that may sound like a completely subjective matter. But this is a gut
feeling that is based on history and philosophy. The history of failures
to settle CH we know; moreover, we know the Levy and Solovay result, which
apparently blocks a vast mass of proposed solutions. The philosophy is also
familiar, but that's more what underlies my gut feeling than the history.
If CH is a definite proposition, what is it about? The standard answer is
the one given by Goedel's platonistic philosophy. In this case, all it
requires us to believe in is the independent existence of the totality of
subsets of the totality of subsets of N, and the totality of functions
between these sets. This I reject as completely implausible metaphysics.
OK, that may be subjective, too, but at a more basic level. If I am not
mistaken, a number of the people working in set theory also reject this
naive platonism, but still take set-theoretical questions like CH
seriously. (I won't try to name names, in order not to mis-represent
their positions; but I have done some canvassing in that respect.) So
then I have to ask what taking CH seriously means. Is it still considered
a definite proposition if one is not a platonist? If so, in what sense?
If not, what will it mean to settle CH?
If I understand him properly, Bill Tait had another way of saying what it
would mean to take CH seriously: let's forget all this silly business
about the ontology of set theory, and just look at "what's true in set
theory". That sort of makes it look like a work of fiction: what's true
in "Hamlet" or "Don Quixote" or "War and Peace"? Well, we know lots of
things that are true in each of these, and things that are false. But
there are also a lot of unsettled questions. I can buy the story version
of set theory, as I would buy the story version of geometry: that's a
story about a universe in which there are perfectly fine points and
perfectly straight lines, etc., etc. In each story, we can go a long way
on very little in the way of characters and plot. But then we come to the
places that the story leaves undetermined. We do feel in mathematics that
the stories, if that's what they are, are less arbitrary than works of
fiction. That's because the kinds of objects these are supposed to be
about are refined to have a minimum few characteristics, and then one has
significantly fewer options as to what to tell about them.
A sociological question: are there mathematicians working on CH the way
mathematicians have been working on RH or twin-prime conjecture or P = NP,
or Schanuel's conjecture? My impression, admittedly as an outsider, that
the work of the last 30-35 years that Steel talks about, is on how the
statement CH relates to other propositions of varying degrees of formal
plausibility, within or relative to axiomatic set theory. That's a very
different kind of enterprise. I venture to guess that none of the
mathematicians working on RH etc. do so in an axiomatic framework or even
with any axioms for mathematics in mind.
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